Problem: $ A = \left[\begin{array}{rr}0 & -1 \\ 0 & 1\end{array}\right]$ $ C = \left[\begin{array}{rrr}0 & 2 & 0 \\ -2 & -1 & -1\end{array}\right]$ What is $ A C$ ?
Answer: Because $ A$ has dimensions $(2\times2)$ and $ C$ has dimensions $(2\times3)$ , the answer matrix will have dimensions $(2\times3)$ $ A C = \left[\begin{array}{rr}{0} & {-1} \\ {0} & {1}\end{array}\right] \left[\begin{array}{rrr}{0} & \color{#DF0030}{2} & \color{#9D38BD}{0} \\ {-2} & \color{#DF0030}{-1} & \color{#9D38BD}{-1}\end{array}\right] = \left[\begin{array}{rrr}? & ? & ? \\ ? & ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ A$ , with the corresponding elements in column $j$ of the second matrix, $ C$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ A$ with the first element in ${\text{column }1}$ of $ C$ , then multiply the second element in ${\text{row }1}$ of $ A$ with the second element in ${\text{column }1}$ of $ C$ , and so on. Add the products together. $ \left[\begin{array}{rrr}{0}\cdot{0}+{-1}\cdot{-2} & ? & ? \\ ? & ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ A$ with the corresponding elements in ${\text{column }1}$ of $ C$ and add the products together. $ \left[\begin{array}{rrr}{0}\cdot{0}+{-1}\cdot{-2} & ? & ? \\ {0}\cdot{0}+{1}\cdot{-2} & ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ A$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ C$ and add the products together. $ \left[\begin{array}{rrr}{0}\cdot{0}+{-1}\cdot{-2} & {0}\cdot\color{#DF0030}{2}+{-1}\cdot\color{#DF0030}{-1} & ? \\ {0}\cdot{0}+{1}\cdot{-2} & ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rrr}{0}\cdot{0}+{-1}\cdot{-2} & {0}\cdot\color{#DF0030}{2}+{-1}\cdot\color{#DF0030}{-1} & {0}\cdot\color{#9D38BD}{0}+{-1}\cdot\color{#9D38BD}{-1} \\ {0}\cdot{0}+{1}\cdot{-2} & {0}\cdot\color{#DF0030}{2}+{1}\cdot\color{#DF0030}{-1} & {0}\cdot\color{#9D38BD}{0}+{1}\cdot\color{#9D38BD}{-1}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rrr}2 & 1 & 1 \\ -2 & -1 & -1\end{array}\right] $